Almgren-Pitts Min-Max Theory is a method of constructing minimal hypersurfaces in Riemannian manifolds. In the last few years a number of long-standing open problems in Geometry, Geometric Analysis and 3-manifold Topology have been solved using this method. I will explain the main ideas and challenges in Min-Max Theory with an emphasis on its quantitative aspect: what quantitative information about the geometry and topology of minimal hypersurfaces can be extracted from the theory?

Almgren-Pitts Min-Max Theory is a method of constructing minimal hypersurfaces in Riemannian manifolds. In the last few years a number of long-standing open problems in Geometry, Geometric Analysis and 3-manifold Topology have been solved using this method. I will explain the main ideas and challenges in Min-Max Theory with an emphasis on its quantitative aspect: what quantitative information about the geometry and topology of minimal hypersurfaces can be extracted from the theory?

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===November 21:Michael Kemeny (Stanford)===

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Title: The equations defining curves and moduli spaces

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Abstract:

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A projective variety is a subset of projective space defined by polynomial equations. One of the oldest problems in

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algebraic geometry is to give a qualitative description of the equations defining a variety, together with

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the relations amongst them. When the variety is an algebraic curve (or Riemann surface), several conjectures

+

made since the 80s give a fairly good picture of what we should expect. I will describe a new variational approach to these conjectures,

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which reduces the problem to studying cycles on Hurwitz space or on the moduli space of curves.

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===December 5: Ryan Hynd (U Penn)===

===December 5: Ryan Hynd (U Penn)===

Revision as of 15:13, 17 November 2017

Mathematics Colloquium

All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.

Fall Abstracts

September 8: Tess Anderson (Madison)

Title: A Spherical Maximal Function along the Primes

Abstract: Many problems at the interface of analysis and number theory involve showing that the primes, though deterministic, exhibit random behavior. The Green-Tao theorem stating that the primes contain infinitely long arithmetic progressions is one such example. In this talk, we show that prime vectors equidistribute on the sphere in the same manner as a random set of integer vectors would be expected to. We further quantify this with explicit bounds for naturally occurring maximal functions, which connects classical tools from harmonic analysis with analytic number theory. This is joint work with Cook, Hughes, and Kumchev.

September 22: Jaeyoung Byeon (KAIST)

Title: Patterns formation for elliptic systems with large interaction forces

Abstract: Nonlinear elliptic systems arising from nonlinear Schroedinger systems have simple looking reaction terms. The corresponding energy for the reaction terms can be expressed as quadratic forms in terms of density functions. The i, j-th entry of the matrix for the quadratic form represents the interaction force between the components i and j of the system. If the sign of an entry is positive, the force between the two components is attractive; on the other hand, if it is negative, it is repulsive. When the interaction forces between different components are large, the network structure of attraction and repulsion between components might produce several interesting patterns for solutions. As a starting point to study the general pattern formation structure for systems with a large number of components, I will first discuss the simple case of 2-component systems, and then the much more complex case of 3-component systems.

October 6: Jonathan Hauenstein (Notre Dame)

Title: Real solutions of polynomial equations

Abstract: Systems of nonlinear polynomial equations arise frequently in applications with the set of real solutions typically corresponding to physically meaningful solutions. Efficient algorithms for computing real solutions are designed by exploiting structure arising from the application. This talk will highlight some of these algorithms for various applications such as solving steady-state problems of hyperbolic conservation laws, solving semidefinite programs, and computing all steady-state solutions of the Kuramoto model.

October 13: Tomoko Kitagawa (Berkeley)

Title: A Global History of Mathematics from 1650 to 2017

Abstract: This is a talk on the global history of mathematics. We will first focus on France by revisiting some of the conversations between Blaise Pascal (1623–1662) and Pierre de Fermat (1607–1665). These two “mathematicians” discussed ways of calculating the possibility of winning a gamble and exchanged their opinions on geometry. However, what about the rest of the world? We will embark on a long oceanic voyage to get to East Asia and uncover the unexpected consequences of blending foreign mathematical knowledge into domestic intelligence, which was occurring concurrently in Beijing and Kyoto. How did mathematicians and scientists contribute to the expansion of knowledge? What lessons do we learn from their experiences?

October 20: Pierre Germain (Courant, NYU)

Title: Stability of the Couette flow in the Euler and Navier-Stokes equations

Abstract: I will discuss the question of the (asymptotic) stability of the Couette flow in Euler and Navier-Stokes. The Couette flow is the simplest nontrivial stationary flow, and the first one for which this question can be fully answered. The answer involves the mathematical understanding of important physical phenomena such as inviscid damping and enhanced dissipation. I will present recent results in dimension 2 (Bedrossian-Masmoudi) and dimension 3 (Bedrossian-Germain-Masmoudi).

October 27: Stefanie Petermichl (Toulouse)

Title: Higher order Journé commutators

Abstract: We consider questions that stem from operator theory via Hankel and
Toeplitz forms and target (weak) factorisation of Hardy spaces. In
more basic terms, let us consider a function on the unit circle in its
Fourier representation. Let P_+ denote the projection onto
non-negative and P_- onto negative frequencies. Let b denote
multiplication by the symbol function b. It is a classical theorem by
Nehari that the composed operator P_+ b P_- is bounded on L^2 if and
only if b is in an appropriate space of functions of bounded mean
oscillation. The necessity makes use of a classical factorisation
theorem of complex function theory on the disk. This type of question
can be reformulated in terms of commutators [b,H]=bH-Hb with the
Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such
as in the real variable setting, in the multi-parameter setting or
other, these classifications can be very difficult.

Such lines were begun by Coifman, Rochberg, Weiss (real variables) and
by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of
spaces of bounded mean oscillation via L^p boundedness of commutators.
We present here an endpoint to this theory, bringing all such
characterisation results under one roof.

The tools used go deep into modern advances in dyadic harmonic
analysis, while preserving the Ansatz from classical operator theory.

November 1: Shaoming Guo (Indiana)

Title: Parsell-Vinogradov systems in higher dimensions

Abstract:
I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions.
Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed.
Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.

November 17:Yevgeny Liokumovich (MIT)

Title: Recent progress in Min-Max Theory

Abstract:
Almgren-Pitts Min-Max Theory is a method of constructing minimal hypersurfaces in Riemannian manifolds. In the last few years a number of long-standing open problems in Geometry, Geometric Analysis and 3-manifold Topology have been solved using this method. I will explain the main ideas and challenges in Min-Max Theory with an emphasis on its quantitative aspect: what quantitative information about the geometry and topology of minimal hypersurfaces can be extracted from the theory?

November 21:Michael Kemeny (Stanford)

Title: The equations defining curves and moduli spaces

Abstract:
A projective variety is a subset of projective space defined by polynomial equations. One of the oldest problems in
algebraic geometry is to give a qualitative description of the equations defining a variety, together with
the relations amongst them. When the variety is an algebraic curve (or Riemann surface), several conjectures
made since the 80s give a fairly good picture of what we should expect. I will describe a new variational approach to these conjectures,
which reduces the problem to studying cycles on Hurwitz space or on the moduli space of curves.

December 5: Ryan Hynd (U Penn)

Title: TBA.

December 11: Connor Mooney (ETH Zurich)

Title: Finite time blowup for parabolic systems in the plane

Abstract:
Hilbert's 19th problem asks about the smoothness of solutions to nonlinear elliptic PDE that arise in the calculus of variations. This problem leads naturally to the question of continuity for solutions to linear elliptic and parabolic systems with measurable coefficients. We will first discuss some classical results on this topic, including Morrey's result that solutions to linear elliptic systems in two dimensions are continuous. We will then discuss surprising recent examples of finite time blowup from smooth data for linear parabolic systems in two dimensions, and important open problems.

December 19: Alex Wright (Stanford)

Title: Dynamics, geometry, and the moduli space of Riemann surfaces

Abstract: The moduli space of Riemann surfaces of fixed genus is one of the hubs of modern mathematics and physics. We will tell the story of how simple sounding problems about polygons, some of which arose as toy models in physics, became intertwined with problems about the geometry of moduli space, and how the study of these problems in Teichmuller dynamics lead to connections with homogeneous spaces, algebraic geometry, dynamics, and other areas. The talk will mention joint works with Alex Eskin, Simion Filip, Curtis McMullen, Maryam Mirzakhani, and Ronen Mukamel.